Convex integer optimization by constantly many linear counterparts

In this article we study convex integer maximization problems with composite objective functions of the form f(Wx)f(Wx), where f   is a convex function on RdRd and W   is a d×nd×n matrix with small or binary entries, over finite sets S⊂ZnS⊂Zn of integer points presented by an oracle or by linear inequalities.Continuing the line of research advanced by Uri Rothblum and his colleagues on edge-directions, we introduce here the notion of edge complexity of S  , and use it to establish polynomial and constant upper bounds on the number of vertices of the projection conv(WS)conv(WS) and on the number of linear optimization counterparts needed to solve the above convex problem.Two typical consequences are the following. First, for any d  , there is a constant m(d)m(d) such that the maximum number of vertices of the projection of any matroid S⊂{0,1}nS⊂{0,1}n by any binary d×nd×n matrix W   is m(d)m(d) regardless of n and S  ; and the convex matroid problem reduces to m(d)m(d) greedily solvable linear counterparts. In particular, m(2)=8m(2)=8. Second, for any d,l,md,l,m, there is a constant t(d;l,m)t(d;l,m) such that the maximum number of vertices of the projection of any three-index l×m×nl×m×n transportation polytope for any n   by any binary d×(l×m×n)d×(l×m×n) matrix W   is t(d;l,m)t(d;l,m); and the convex three-index transportation problem reduces to t(d;l,m)t(d;l,m) linear counterparts solvable in polynomial time.